Neyman criterion

In mathematical statistics, the Neyman criterion is a criterion for the sufficiency of σ-algebras and sufficiency of statistics in statistical models with dominated distribution classes . The Neyman criterion is derived from the Halmos-Savage theorem , but it is easier to apply than this. The Neyman criterion is thus one of the most common criteria for checking whether an image compresses data without loss of information.

It is named after Jerzy Neyman .

statement

For σ-algebras

A statistical model with a dominated distribution class , which is dominated by, and a sub-σ-algebra of . ${\displaystyle (\Omega ,{\mathcal {A}},{\mathcal {P}})}$ ${\displaystyle {\mathcal {P}}}$ ${\displaystyle \nu }$ ${\displaystyle {\mathcal {S}}}$ ${\displaystyle {\mathcal {A}}}$

Then suffizient if and only if a - measurable function exists for each one -measurable function exists so that ${\displaystyle {\mathcal {S}}}$ ${\displaystyle {\mathcal {A}}-{\mathcal {B}}([0,\infty ))}$ ${\displaystyle h}$ ${\displaystyle P\in {\mathcal {P}}}$ ${\displaystyle {\mathcal {S}}-{\mathcal {B}}([0,\infty ))}$ ${\displaystyle f_{P}}$

${\displaystyle {\frac {\mathrm {d} P}{\mathrm {d} \nu }}(x)=h(x)\cdot f_{P}(x)}$

applies except for a zero set. It is the Radon Nikodým derivation of respect . ${\displaystyle \nu }$ ${\displaystyle {\tfrac {\mathrm {d} P}{\mathrm {d} \nu }}(x)}$ ${\displaystyle P}$ ${\displaystyle \nu }$

For statistics

A statistic is subject to the same conditions as above

${\displaystyle T:(\Omega ,{\mathcal {A}})\to (\Omega ^{*},{\mathcal {A}}^{*})}$

sufficient if and only if a measurable function exists and for each a measurable function exists such that ${\displaystyle {\mathcal {A}}-{\mathcal {B}}([0,\infty ))}$ ${\displaystyle h}$ ${\displaystyle P\in {\mathcal {P}}}$ ${\displaystyle \Omega ^{*}-{\mathcal {B}}([0,\infty ))}$ ${\displaystyle f_{P}}$

${\displaystyle {\frac {\mathrm {d} P}{\mathrm {d} \nu }}(x)=h(x)\cdot f_{P}(T(x))}$

applies except for a zero set. This follows from the factoring lemma and the fact that there is a sufficient statistic if and only if there is a sufficient σ-algebra. ${\displaystyle \nu }$ ${\displaystyle T}$ ${\displaystyle \sigma (T)}$

Example: sufficiency of the exponential family

By definition has the exponential respect each the density function ${\displaystyle {\mathcal {P}}}$ ${\displaystyle \mu }$ ${\displaystyle P\in {\mathcal {P}}}$

${\displaystyle f_{\vartheta }(x)={\frac {\mathrm {d} P}{\mathrm {d} \mu }}(x)=C(\vartheta )h(x)\exp(\langle Q(\vartheta );T(x)\rangle )}$

But this is exactly the decomposition required above. and are already correct, you just bet ${\displaystyle h}$ ${\displaystyle T}$

${\displaystyle f_{P}(y)={\tilde {f}}_{\vartheta }(y)=C(\vartheta )\exp(\langle Q(\vartheta );y\rangle )}$

to show that is a sufficient statistic for the exponential family. ${\displaystyle T}$

literature

• Ludger Rüschendorf: Mathematical Statistics . Springer Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-41996-6 , doi : 10.1007 / 978-3-642-41997-3 .